Transitions metal dichalcogenides (TMDs) are direct gap semiconductors in the monolayer (ML) limit with fascinating optical and spin-valley properties. The strong optical absorption of up to 20% for a single ML is governed by excitons, electron-hole pairs bound by Coulomb attraction. Excited exciton states in MoSe2 and MoTe2 monolayers have so far been elusive because of their low oscillator strength and strong inhomogeneous broadening. Here, we show that encapsulation in hexagonal boron nitride results in an emission line width of the A:1s exciton below 1.5 meV and 3 meV in our MoSe2 and MoTe2 monolayer samples, respectively. This allows us to investigate the excited exciton states by photoluminescence upconversion spectroscopy for both monolayer materials. The excitation laser is tuned into resonance with the A:1s transition, and we observe emission of excited exciton states up to 200 meV above the laser energy. We demonstrate bias control of the efficiency of this nonlinear optical process. We discuss the origin of the upconversion effect. Our model calculations suggest an exciton-exciton (Auger) scattering mechanism specific to TMD MLs involving an excited conduction band, thus generating high-energy excitons with small wave vectors. The optical transitions are further investigated by white light reflectivity, photoluminescence excitation, and resonant Raman scattering, confirming their origin as excited excitonic states in monolayer thin semiconductors.